Learning Topological Quantum Phases from Limited Subsystems

Abstract

Characterizing quantum topological phases requires measuring non-local string order parameters, demanding access to the full system, which is often experimentally unfeasible. In this work, we introduce a data-efficient supervised learning framework that circumvents this limitation by recognizing quantum phases from small subsystems. Our protocol utilizes a quantum kernel constructed from the reduced density matrices of these subsystems, which can be efficiently estimated experimentally. We benchmark our framework with the classification of the phase diagrams of two spin models on one-dimensional lattices, namely the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain. Remarkably, our approach achieves high accuracy in phase classification when operations are limited to as few as one to four sites, and it also generalizes to longer chains even when trained on moderate system sizes. These findings demonstrate that local reduced density matrices preserve vital signatures of global topological phases, offering a practical route to characterize rich phase diagrams of quantum many-body systems.

The paper asks whether phases in one-dimensional spin systems, including symmetry-protected topological (SPT) phases, can be recognized when measurements are restricted to a small subsystem rather than nonlocal string observables. It studies supervised classification within known phase diagrams, using local information to distinguish phases without directly measuring their topological order. The method feeds the reduced-density-matrix overlap $K(x,x')=\mathrm{Tr}[\rho_A(x)\rho_A(x')]$ into a classical support-vector machine. Training examples are ground states labeled using established phase diagrams; reduced states are calculated from matrix-product-state simulations. Benchmarks cover the generalized cluster-Ising spin-$1/2$ chain and an anisotropic spin-$1$ Haldane chain. Subsystems comprise one to four contiguous sites, located either centrally or at an edge. For the cluster-Ising model, three-site central blocks identify the phases accurately, with four sites improving their resolution. A single central site separates the SPT region from alternatives but fails to distinguish antiferromagnetic and paramagnetic regions. For the Haldane model, even one central site reproduces much of the phase structure, while identifying Néel phases improves with block size. Central blocks generally outperform edge blocks. Training on chains of length $31$ and testing on length $51$ gives results comparable to training and testing at length $51$. The stated qualification is that recognition relies on prior phase labels and model-specific differences between local reduced states: single-site data do not directly capture nonlocal topological correlations. Experimental overlap-estimation methods are discussed, but the reported demonstrations are numerical.